Given n random variables $X_1, \ldots , X_n$ taken from known distributions, a gambler observes their realizations in this order, and needs to select one of them, immediately after it is being observed, so that its value is as high as possible. The classical prophet inequality shows a strategy that guarantees a value at least half (in expectation) of that an omniscience prophet that picks the maximum, and this ratio is tight. Esfandiari, Hajiaghayi, Liaghat, and Monemizadeh introduced a variant of the prophet inequality, the prophet secretary problem in [1]. The difference being that that the realizations arrive at a random permutation order, and not an adversarial order. Esfandiari et al. gave a simple $1-1/e \approx 0.632$ competitive algorithm for the problem. This was later improved in a surprising result by Azar, Chiplunkar and Kaplan [2] into a $1-1/e + 1/400 \approx 0.634$ competitive algorithm. In a subsequent result, Correa, Saona, and Ziliotto [3] took a systematic approach, introducing blind strategies, and gave an improved $0.669$ competitive algorithm. Since then, there has been no improvements on the lower bounds. Meanwhile, current upper bounds show that no algorithm can achieve a competitive ratio better than $0.7235$ [4]. In this paper, we give a $0.6724$-competitive algorithm for the prophet secretary problem. The algorithm follows blind strategies introduced by [3] but has a technical difference. We do this by re-interpretting the blind strategies, framing them as Poissonization strategies. We break the non-iid random variables into iid shards and argue about the competitive ratio in terms of events on shards. This gives significantly simpler and direct proofs, in addition to a tighter analysis on the competitive ratio. The analysis might be of independent interest for similar problems such as the prophet inequality with order-selection

翻译：给定来自已知分布的 $n$ 个随机变量 $X_1, \ldots , X_n$，一个赌徒按顺序观察它们的实现值，并需要在观察后立即选择一个，使其值尽可能大。经典先知不等式给出一种策略，保证其期望值至少达到全知先知（会选择最大值）的一半，且该比率是紧的。Esfandiari、Hajiaghayi、Liaghat 和 Monemizadeh 在文献 [1] 中引入了先知不等式的一个变体——先知秘书问题。其区别在于实现值以随机排列顺序到达，而非对抗性顺序。Esfandiari 等人为该问题提出了一种简单的 $1-1/e \approx 0.632$ 竞争比算法。随后，Azar、Chiplunkar 和 Kaplan 在文献 [2] 中通过一项令人惊讶的结果将其改进为 $1-1/e + 1/400 \approx 0.634$ 竞争比算法。在后续工作中，Correa、Saona 和 Ziliotto 在文献 [3] 中采用系统方法，引入了盲策略，并给出了改进的 $0.669$ 竞争比算法。自那以后，下界方面未再有改进。同时，当前上界表明，没有算法能实现优于 $0.7235$ 的竞争比 [4]。在本文中，我们为先知秘书问题给出一个 $0.6724$ 竞争比算法。该算法遵循文献 [3] 引入的盲策略，但在技术上有所差异。我们通过重新解释盲策略，将其框架化为泊松化策略来实现这一点。我们将非独立同分布的随机变量分解成独立同分布的分片，并依据分片上的事件来论证竞争比。除了对竞争比进行更紧的分析外，这还提供了显著更简单、更直接的证明。该分析可能对类似问题（如具有顺序选择的先知不等式）具有独立意义。